Abstract
In this paper, the mathematical formulation for a quadratic optimal control problem governed by a linear hyperbolic integro-differential equation is established. We first show the existence and regularity for the solution of the optimal control problem. The finite element approximation is based on the optimality conditions, which are also derived. Then the a priori error estimates for its finite element approximation are obtained with the optimal convergence order. Furthermore some numerical tests are presented to verify the theoretical results.
Highlights
The distributed optimal control problem has been a classic research topic in the discipline of applied mathematics
Since it is normally difficult to obtain a closed form solution, finite element approximations of optimal control problems governed by partial differential equations have been extensively studied in the literature
The purpose of this paper is to investigate the weak formulation of the optimal control problem governed by integro-differential equations of hyperbolic type, and its finite element approximation
Summary
The distributed optimal control problem has been a classic research topic in the discipline of applied mathematics. For optimal control problems governed by classic linear PDEs such as elliptic, parabolic and hyperbolic equations, the existence and the optimality conditions are well known, see [ ]. Their finite element approximation and a priori error estimates were established long ago, for example, see [ – , ]. The purpose of this paper is to investigate the weak formulation of the optimal control problem governed by integro-differential equations of hyperbolic type, and its finite element approximation. In Section , we establish the optimal a priori error estimates for the finite element approximation of the control problem. We present some numerical tests, which illustrate the theoretical results
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